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MINIMALITY OF THE EHRENFEST WIND-TREE
MODEL
Alba Málaga Sabogal, Serge Troubetzkoy
To cite this version:
Alba Málaga Sabogal, Serge Troubetzkoy. MINIMALITY OF THE EHRENFEST WIND-TREE
MODEL. Journal of modern dynamics, American Institute of Mathematical Sciences, 2016,
�10.3934/jmd.2016.10.209�. �hal-01158924v2�
ALBA MÁLAGA SABOGAL AND SERGE TROUBETZKOY
Abstract. We consider aperiodic wind-tree models, and show that for a generic (in the sense of Baire) configuration the wind-tree dynamics is minimal in almost all directions, and has a dense set of periodic points.
1. Introduction
In 1912 Paul et Tatyana Ehrenfest proposed the wind-tree model in order to interpret the ergodic hypothesis of Boltzmann [EhEh]. In the Ehrenfest wind-tree model, a point particle (the “wind”) moves freely on the plane and collides with the usual law of geometric optics with irregularly placed identical square scatterers (the “trees”). Nowadays we would say “randomly placed”, but the notion of “ran-domness” was not made precise, in fact it would have been impossible to do so before Kolmogorov laid the foundations of probability theory in the 1930s. The wind-tree model has been intensively studied by physicists, see for example [BiRo], [DeCoVB], [Ga], [HaCo], [VBHa], [WoLa] and the references therein.
From the mathematical rigorous point of view, there have been many recent results about the dynamical properties of a periodic version of wind-tree mod-els, scatterers are identical square obstacles one obstacle centered at each lattice point. The periodic wind-tree model has been shown to be recurrent ([HaWe], [HuLeTr],[AvHu]), to have abnormal diffusion ([DeHuLe],[De]), and to have an ab-scence of egodicity in almost every direction ([FrUl]). Periodic wind-tree models naturally yield infinite periodic translation surfaces, ergodicity in almost every di-rection for such surfaces have been obtained only in a few situations [HoHuWe], [HuWe], [RaTr].
On the other hand for randomly placed obstacles, from the mathematically rig-orous point of view, up to know it has only been shown that if at each point of the lattice Z2 we either center a square obstacle of fixed size or omit it in a random
way, then the generic in the sense of Baire wind-tree model is recurrent and has a dense set of periodic points ([Tr1]).
In this article we continue the study of the Baire generic properties of wind-tree models. We study a random version of the wind-tree model: the plane is tiled by one by one cells with corners on the lattice Z2, in each cell we place a square tree
of a fixed size with the center chosen randomly. Our main result is that for the generic in the sense of Baire tree model, for almost all directions the wind-tree model is minimal, in stark contrast to the situation for the periodic wind-wind-tree
model which can not have a minimal direction.1 This result can be viewed as a
topological version of the Ehrenfests question.
1K. Frączek explained to us that this follows from arguments close to those in the article [Be]. 1
The method of proof is by approximation by finite wind-tree models where the dynamics is well understood. There is a long history of proving results about billiard dynamics by approximation which began with the article of Katok and Zemlyakov [KaZe]. This method was used in several of the results on wind-tree models mentioned above [HuLeTr],[AvHu],[Tr1], see [Tr] for a survey of some other usages in billiards. The idea of approximating infinite measure systems by compact systems was first studied in [MS].
The structure of the article is as follows, in Section 2 we give formal statements of our results. In Section 3 we collect the notation necessary for our setup. In Section 4, 5 and 6 are devoted to the proof of different parts of the main theorem. Our proofs hold in a more general setting than the one described above, for example we can vary the size of the square, or use certain other polygonal trees. We discuss such extensions of our result in Section 7. Finally in Appendix A we discuss the relationship between the usual convention on the orbit of singular points for interval exchange transformations and for polygonal billiards, these conventions are not the same. Since we are studying minimality in this article a careful comparison is made, and certain known results are reproved for a class of maps we call eligible. In particular the IETs arising from billiards with the billiard convention for orbits arriving at corners of the polygon are eligible maps.
2. Statements of Results
We consider the plane R2tiled by one by one closed square cells with corners on
the lattice Z2. Fix r
∈ [1/4, 1/2). We consider the set of 2r by 2r squares, with vertical and horizontal sides, centered at (a, b) contained in the unit cell [0, 1]2, this
set is naturally parametrized by
A := {t = (a, b) : r ≤ a ≤ 1 − r, r ≤ b ≤ 1 − r}
with the usual topology inherited from R2. Our parameter space is AZ2
with the product topology. It is a Baire space. Each parameter g = (ai,j, bi,j)(i,j)∈Z2∈ AZ
2
corresponds to a wind-tree table in the plane in the following manner: the tree inside the cell corresponding to the lattice point (i, j) ∈ Z2 is a 2r by 2r square
with center at position (ai,j, bi,j) + (i, j). The wind-tree table Bg is the plane R2
with the interiors of the union of these trees removed. Note that trees can intersect only at the boundary of cells.
Fix a direction θ ∈ S1. The billiard flow in the direction θ is the free motion
on the interior of Bgwith elastic collision from the boundary of Bg(the boundary
of the union of the trees). The billiard map Tθg in the direction θ on the table is the first return to the boundary. If the flow orbit arrives at a corner of the table, the collision is not well defined, and we choose not to define the billiard map, i.e. the orbit stops at the last collision with the boundary before reaching the corner; also backwards orbits starting at a corner of a tree are not defined, but forward orbits starting at a corner are defined. Once launched in the direction θ, the billiard direction can only achieve four directions{±θ, ±(θ−π)}; thus the phase space Ωgθof the billiard map Tθgis a subset of the cartesian product of the boundary with these four directions. It contains precisely the pairs (s, φ) such that at s the direction φ points to the interior of the table, i.e. away from the trees. The billiard map will be called minimal if the orbit or every point is dense.
The set of periodic points is called locally dense if there exists a Gδ-subset of
the boundary which is of full measure, such that for every s in this set, there is a dense set of inner-pointing directions θ ∈ S1 for which (s, θ) is periodic. We call
a forward (resp. backward) Tθg-orbit a forward (resp. backward) escape orbit if it visits any compact set only a finite number of times.
Theorem 2.1. There is a dense Gδ set of parametersG such that for each g ∈ G:
i) for a dense-Gδ set of full measure of θ the billiard map Tθg is minimal and has
forward and backward escape orbits,
ii) the map Tg has a dense set of periodic points,
iii) if r is rational, then the map Tg has a locally dense set of periodic points,
iv) no two trees intersect.
All the sets mentioned in the theorem depend on the fixed parameter r. From our definition of minimality we conclude
Corollary 2.2. The backwards orbit of any forward escape orbit is dense (and vice
versa) for each g∈ G.
Corollary 2.3. The billiard flow on the wind-tree table is also minimal for each
g∈ G.
We would like to point out that there is an old theorem of Gottshalk that (a stronger version of) minimality is impossible in locally compact spaces [Go, Theo-rem B]; more precisely for a homeomorphism of a locally compact metric space X, if the forward orbit of every point y∈ Y is dense in Y , then Y is compact. This re-sult does not apply directly to our situation: our map is not a homeomorphism, the dynamics is not defined everywhere, and where it is not defined it is discontinuous. There is a standard way of changing the topology to make the map a homeomor-phism (this construction is well described in the context of interval exchanges in
[MMY, Section 2.1.2]). For any wind tree table g ∈ AZ2, including the periodic
ones the topology obtained from Ωgθ will be locally compact, thus Gottshalks result apply, the wind tree model can never be forward minimal. In fact, in Theorem 2.1.i) we construct examples of escape orbits.
3. Notations and preparatory remarks
As already mentioned in previous section the billiard map Tθg is not defined at a point whose next collision is with a corner, and the inverse billiard map (Tθg)−1
is not defined at a corner. In the world of billiards or (flat surfaces), a saddle connection is a flow-orbit going from a corner of a tree to some corner (maybe the same one). Because of the above convention, for the map there is a saddle connection starting at a point x if, for some k ≥ 0, Tk(x) is defined but Tk+1(x)
and T−1(x) are not defined; then the saddle connection is the orbit
{x, T x, T2(x), . . . , Tk(x)
}.
A direction θ is called exceptional if there exists a saddle connection for Tθg. As there are countably many corners, there are at most countable many saddle connections and thus at most countably many exceptional directions.
For any positive integer N , we define RN to be the closed rhombus (square)
{(x, y) : |x|+|y| ≤ N +1
2} and we define then Ω g
θ,N to be Ω g
Figure 1. A 2-ringed configuration.
Figure 2. A small
per-turbation.
Let EN the set of pairs (i, j) so that the interior of the (i, j)-th cell is contained in
RN, and let RN be the interior of the union of the closed cells indexed by EN. Let
us also define Ωgθ,N to be Ωgθ∩ (RN × {±θ, ±(θ − π)}).
Suppose that N is an integer satisfying N ≥ 2. We will call a parameter
N-tactful if for each cell inside the rhombus RN, the corresponding tree is contained in the interior of its cell. We will call an N -tactful parameter N -ringed, if the boundary of RN is completely covered by trees. We call a parameter tactful if it is N -tactful for all N .
For N -ringed parameters there is a compact connected rational billiard table RN ∩ Bf, called the N -ringed table, contained in the rhombus R
N (see Figure 1).
The corresponding phase space is Ωf
θ,N. It contains Ω f
θ,N which is compact for any
N -tactful parameter. A direction θ is called (f, N )-exceptional if there is a saddle connection inside Ωfθ,N.
There are at most countably many exceptional directions, and for all non-exceptional directions, Ωf
θ,N is a minimal set for the billiard map T f
θ. (We reprove
this result in our context in Corollary A.6 of the Appendix.)
We need to describe Ωgθ,N more concretely for any N -tactful g. Note that if the tree is contained in the interior of a cell then if s is a corner of this tree there are three directions pointing to the interior of the table, while for all other s there are only two such directions (see Figure 3). Intersecting trees can have slightly different behavior, but we do not need to describe it since they will not occur in our proof.
We think of the contribution of each tree to Ωgθ,N as the union of four closed
intervals indexed by φ ∈ {±θ, ±(θ − π)}, each of these intervals corresponds to
the cartesian product of the two intersecting sides of the tree with a fixed inner pointing direction φ (see Figure 3) (as before the word inner means pointing into the table, so away from the tree). In the proof we will think of each of these intervals as I = [0, 2√2r] since it corresponds to the diagonal (of length 2√2r) of the tree centered at (a, b). Note that the billiard map in a fixed direction has a natural invariant measure. Let p be the arc-length parameter on I, then I = I1∪ I2 where
each Ii is an interval and the invariant measure is of the form Kid p on Iiwith Ki
an explicit constant. We stick to the use of I in order to avoid manipulating the constants Ki (which are direction dependent) all the time.
Figure 3. The phase space of one tree.
Figure 4. The phase space is
the disjoint union of four closed oriented “intervals”.
To make our map orientation preserving we choose the orientation of these in-tervals in the following way; use the clockwise orientation inherited from the tree for φ∈ {θ, θ − π} and the counterclockwise orientation of the other two values of φ. In particular this parametrization does not depend on the angle φ. (See figures 3,4). Despite that fact that these “intervals” come naturally as subsets of R2, we
will think of Ωgθ,N and Ωgθ as formal disjoint union of one-dimensional intervals. For any N -tactful g, let JNg be the collection of all the intervals as described
arising from the trees in RN. Note that the trees straddling the rhombus do not
contribute to this collection.
For each tree t∈ A let U(t, ε) be the the standard ε-neighborhood in R2
inter-sected with the interior of A in R2. For any parameter g = (t
i,j)∈ AZ
2
, consider the open cylinder set UN(g, ε) =Q(i,j)∈ENU (ti,j, ε).
4. Proof of Minimality in Theorem 2.1
Proof. In the Appendix we study a class of maps called eligible maps. The proof
of minimality in the theorem is based on Lemma A.1 of the Appendix which gives a necessary and sufficient condition for the minimality of an eligible map. We start by some remarks on the applicability of this lemma, first of all note that in the proof we will only need to apply Lemma A.1 to maps which are tactful. However these maps are not eligible. Ωgθ is a disjoint union of closed intervals, if we restrict the billiard map to the interior of these intervals it becomes an eligible map and we
can apply the lemma. More precisely we apply Lemma A.1 to the map ˜Tθg which
is the map Tθg restricted to Ωgθ with endpoints of each interval inJNg removed. We call this union of open intervals ˜Ωgθ. Note that Tθg being minimal is equivalent to
˜
Tθg being minimal since the Tθg-orbit of any corner x is the union of {x} with the ˜
Tθg-orbit of Tθg(x).
By Lemma A.1 for any tactful g, the map ˜Tθgbeing minimal is equivalent to the statement: for any interval I ⊂ ˜Ωgθ we haveSk∈Z( ˜Tθg)k(I) covers the whole space
˜
Ωgθ. It is enough to show that this happens for a finite union of iterates of I. More precisely it is enough to show that we have sets Cn⊂ ˜Ωgθsatisfying Cn ⊂ Cn+1and
∪n≥1Cn= ˜Ωgθ , such that ∀n ≥ 1 ∀I ⊂ ˜Ωgθ∃K, L s.t L [ k=K ( ˜Tθg)k(I)⊃ Cn.
Furthermore it suffices to show this for a countable basis of intervals.
By Corollary A.4 in the Appendix for any N -ringed f , any (f, N )-non-exceptional direction θ and any interval I ⊂ ˜Ωfθ,N, there exists K, L such that
(1)
L
[
k=K
( ˜Tθf)k(I)⊃ ˜Ωfθ,N.
Now consider the following perturbation of f , the new configuration g is arbitrary in the cells which do not intersect RN, each tree (ai,j, bi,j) in RN is replaced with
a tree (a′
i,j, b′i,j) which is sufficiently close to (ai,j, bi,j) in such a way that the new
parameter is still N -tactful and the trees in the cells covering the boundary of RN are replaced by close trees in such a way that the configuration is N + 1-tactful (see Figure 2).
The main idea is if Equation (1) holds for an open interval I in ˜Ωfθ,N there exists an ε > 0 such that for all g ∈ UN(f, ε), the Equation (1) still holds for g and I,
namely L [ k=K ˜ Tθgk(I)⊃ ˜Ωgθ,N.
Here we can write the same interval I since there is a natural identification between ˜
Ωgθ,N and ˜Ωfθ,N which will be made explicit in the proof.
We remind that each tree contributes four intervals to the phase space of the wind-tree transformation in a given direction. Each of these intervals is a copy of the interval [0, 2√2r]. Since in ˜Ωgθ,N only the trees contained in RN are contributing,
JNg is a collection of 4 card(EN) = 8N (N− 1) copies of this interval.
For any N -tactful g,IN,θg will be the union of all open coverings
(2) ( √ 2r 2N (i− 1), √ 2r 2N (i + 1) ! ∩ (0, 2√2r) : i = 0, . . . , 2N+1 )
of (0, 2√2r), one such open covering for each copy of the interval [0, 2√2r] which appears in JNg. Note that we have remove the endpoints of the interval [0, 2√2r] because of the discussion at the beginning of the proof of the applicability of the results in the Appendix. ThusIN,θg is a finite collection of open intervals in ˜Ωgθ,N.
Note also that S
NI g
N,θ is a topological basis of ˜Ω g
θ. We will call the endpoints of
intervals inIN,θg dyadic points (the endpoints 0 and 2√2r included).
We will say that there is a *saddle connection starting at a point x∈ Ωgθ if for
some k≥ 0 we have: Tθg
i
(x) is defined and is not a dyadic point for all 0 < i≤ k, and Tθgk+1(x) and Tθg−1(x), if defined, are dyadic points ; then the *saddle
connection is the orbit segment
n
A direction θ is called ((g, N ))-*exceptional if there is a *saddle connection
in-side Ωgθ,N. There are at most countably many *exceptional directions. Any
*exceptional direction is exceptional, thus Equation (1) still holds for any non-*exceptional direction θ, for f an N -ringed configuration.
To prove that the billiard map ˜Tθg in a given direction is minimal, it suffices to show that there exists infinitely many N such that
(3) ∃K, L ∀I ∈ IN,θg
L
[
k=K
( ˜Tθg)k(I)⊃ ˜Ωgθ,N.
For any N -tactful g, letCNg(K, L) be the collection of all the connected
compo-nents of ( ˜Tθg)k(I)∩ ˜Ωg
θ,N where k varies from K to L. These intervals are open
intervals. Each interval I′ in
CNg(K, L) is a connected component of ( ˜T g
θ)k(I) for
some k between K and L, and some I ∈ IN,θg . Thus, for this k, ( ˜Tθg)−k(I′) is an
interval in ˜Ωgθ,N, we will callDNg(K, L) the collection of all such intervals.
Note that ˜Ωgθ,N and ˜Ωfθ,N for every N -ringed parameter f and every N -tactful g are formally identical. In particular this is true for any g in the cylinder set UN(f, ε) (defined at the end of the previous section).
By Baire’s theorem the set of configurations which are tactful is dense since for each N the set of all N -tactful configurations is an open dense set. Thus we can consider a countable dense set of parameters which are N -tactful for all N . By modifying the parameters we can assume that each one is N -ringed for a certain N still maintaining the density. Call this countable dense set {fi}, with fi being
Ni-ringed. We also assume Ni+1> Ni. Suppose εi are strictly positive. Let
G := \
m≥1
[
i≥m
UNi(fi, εi).
Clearly G is a dense Gδ. We claim that there is a choice for εi such that every
parameter inG gives rise to a wind tree which is minimal in almost all directions. Fix fi. We already proved that Equation (3) holds for g = fi, N = Ni and θ
any direction which is not (fi, Ni)-exceptional (c.f. Equation (1)). Let Ki= Ki(θ),
Li = Li(θ) be the two integers given by Equation (3). For sake of simplicity, we
will denoteCi:=CNfii(Ki, Li) the collection of intervals in the covering in Equation
(3) and we will denote Di :=DNfii(Ki, Li) the collection defined in the paragraph
after Equation (3). The collection of intervalsCi is an open cover of the open set
˜ Ωfi
θ,Ni. We denote by ∂Ci the set of endpoints in Ω
fi
θ of the intervals inCi.
We describe the set ∂Ci(θ) exactly. Without loss of generality let suppose
Ki(θ) < 0 and Li(θ) > 0. First, let us consider the set Ωfθ,Ni i\ T
fi θ Ki Ωfi θ,Ni which is just the collection of points x whose forward iterate Tfi
θ
k
(x) is not de-fined for some time k ≤ −Ki(θ), and similarly consider Ωfθ,Ni i\ T
fi θ Li Ωfi θ,Ni for backward orbits. Second, let us consider the following sub-collection of the for-ward orbits of corners of trees ˜Tk
θ(x) : x ∈ ∂JNi, 0 ≤ k ≤ Li
, and similarly for backward iterates. Third, consider the iterates of the endpoints of I for times between Ki(θ) and Li(θ). Then ∂Ci(θ) is the restriction of the union of these three
collections to Ωfi
θ,Ni. For each θ this is a finite collection of points. Note that the
is not (fi, Ni)-*exceptional. As we vary the parameters (g, θ), clearly ∂Ci(θ) will
change. Moreover, if the direction θ is a non-(fi, Ni)-*exceptional direction, then
the points in the set change continuously in the following sense: each point in ∂Ci(θ)
has (fi, θ) as a point of continuity.
Let Θibe the set of all directions θ who are not (fi, Ni)-*exceptional. This set is
of measure one since its complement is countable. For every θ∈ Θ the points in the collection ∂Ci(θ) are all distinct. Recall that Ωfθ,Ni i is a union of oriented intervals,
so the intersection of ∂Ci(θ) with any of these intervals has a natural total order
(which is a strict order). These orders induce a strict partial order in ∂Ci(θ).
So, for every fixed θ ∈ Θi it is possible to choose continuously δi(θ) > 0 such
that the strict partial order on ∂Ci(θ′) is preserved for all (g, θ′) in the open set
UNi(fi, δi(θ))× B(θ, δi(θ)) (where B denotes a ball in S
1); and thus, Equation (3)
holds with the same Ki(θ) and Li(θ) for every such (g, θ′).
Furthermore, let us suppose now that Ki(θ) and Li(θ) are optimal for Equation
(3) to hold. By this we mean the−Ki(θ) + Li(θ) is minimal, and then if there are
several choices Ki(θ) is chosen maximal still satisfying Ki(θ) < 0. Then
ΘK,L,M,i:={θ ∈ Θi: Ki(θ)≥ K, Li(θ)≤ L, δi(θ) >
1
M}
is an open set. Since Θi = SK≤0
S
L≥0
S
M≥1ΘK,L,M,i is an increasing union of
ΘK,L,M,iand is of full measure, there exists Ki, Li, Misuch that bΘi:= ΘKi,Li,Mi,i
is an open set of measure larger than N1i. Equation (3) holds with the constants Ki, Lifor every θ∈ bΘi. Thus
Θ := \ N≥1 [ {i≥N } b Θi
is a Gδ-dense set of full measure. Without loss of generality, we suppose Mi is
increasing. If we choose εi= M1i in the definition ofG, then this is a set of minimal
directions for all tables g∈ G.
5. Proof of the existence of escape orbits.
Proof. We will proof the existence of an escape orbit for the parameter set G and
direction set Θ defined in the proof of minimality in Section 4. For any g∈ G and θ∈ Θ, as shown above θ will be a minimal direction of T = Tθg.
Let consider f an N -ringed parameter and ε > 0. Let be N′> N and f′ an N′
-ringed parameter in UN(f, ε). Then, for any ε′> 0, the set UN(f, ε)∩ UN′(f′, ε′) is
non-empty, moreover if ε′ is small enough then U
N′(f′, ε′)⊂ UN(f, ε). Let us now
fix such an ε′, and let θ be a direction which is far for horizontal and vertical in
the following sense: min(| tan(θ)|, | cot(θ)|) ≥ ε′
2r. Then for any g∈ UN′(f
′, ε′) we
have that Ωgθ,N′+1\ Ω g
θ,N′ is visited by any orbit starting in Ω
g
θ,N′ before reaching
Ωg
\ Ωgθ,N′+1 since it makes a collision with one of the squares in the ring. In
particular for any point x∈ Ωgθ,N its orbit cannot escape to Ωg\ Ω g
θ,N′+1without a
collision in the ring of obstacles Λgθ,N′:= Ω
g
θ,N′+1\ Ω g
θ,N′. Similarly in the opposite
way, an orbit that is already outside cannot come inside without a collision on the ring of obstacles.
Let g ∈ G and let us consider an approximating sequence fij such the g ∈
UNij(fij, εij) for all j. For simplicity of notation let Λj = Λ
g
θ,N ij. For any non
horizontal and non vertical direction θ, the condition min(| tan(θ)|, | cot(θ)|) ≥ εij
discussed above is verified for j large enough since εiis a decreasing sequence going
to zero. Thus, we can choose J so large that for any j ≥ J, the εij is sufficiently
small such that for any x ∈ Ωgθ,N
ij the T orbit of x must visit the set Λj before
reaching Λj+1. The same is true in the opposite direction: no orbit can go from
Ωgθ\ Ωgθ,1+Nij+1 to Λj without a collision on Λj+1. (A similar statement holds for
T−1. )
Thus the far away dynamics of T can be understood via the following transfor-mations. For any j > J and any x ∈ Λj, let S+(x) be Tk(x), the first visit to
Λj±1. (Note that this is not a first return map to the union of the Λj and it is not
invertible.) Similarly, S−(x) for any x in Λ
j is T−k(x) where k is minimal such
that T−k(x)
∈ Λj±1. For x∈ Λj, let R+(x) = Tk
′
(x) where k′is the maximal i
≥ 0 so that x, T (x), . . . , Ti(x)
∈ Λj. Similarly we define for x ∈ Λj, R−(x) = T−k(x)
where k is the maximal i≥ 0 so that x, T−1(x), . . . , T−i(x)
∈ Λj.
Where defined, these transformations satisfy:
(4) R − ◦ S+ = S+ = R− ◦ R+ ◦ S+, R+◦ S− = S− = R+◦ R−◦ S−, S− ◦ S+ = R+ and S+ ◦ S− = R−.
Now suppose θ ∈ Θ. Note that θ is not vertical nor horizontal because these
directions are (fi, Ni)-exceptional for every i. Consider the compact set Ωgθ,Mj for
j≥ J. Let
Aj,1:=x∈ Λj: S+(x)∈ Λj+1 .
The set Aj,1 is non-empty since the (forward) T -orbit of any corner of a tree in Λj
is dense in Ωgθ, thus it has to get out of Ωgθ,j+1 and in doing so it is forced to have
a collision in Λj+1. Thus the last time this orbit visits Λj before visiting Λj+1 will
be an element of Aj,1. Now inductively define the set
Aj,n+1:=
n
x∈ Aj,n : ∃k > 0, such that ∀i = 1, 2, . . . , k − 1,
Si(x)6∈ Λj and Skx∈ Λj+n
o .
For each n≥ 1 the set Aj,n is non-empty by a similar reasoning as above. Clearly
Aj,n+1 ⊂ Aj,n, thus since Λj is compact, Bj :=∩n≥1Aj,n is non-empty. We claim
that if x is in this intersection, then x is in all of the Aj,n, and thus the forward
orbit of x never returns to Λj.
Suppose not, then let m := min{n ≥ 1 : x ∈ Aj,n\ Aj,n}. This implies that
for some k, Ti(x) 6∈ Λ
j for i = 1, . . . , k − 1 and Tk(x) is not defined, in fact
Tk(x) would arrive at a corner of a tree of the obstacle ring Λ
j+m. More precisely
chose a sequence (xℓ) ⊂ Aj,m such that xℓ → x and Tk(x) ∈ Λj+m, then y =
limℓ→∞Tk(xℓ). Since our direction θ is non-exceptional, the forward orbit of y is
infinite. Furthermore, we have y∈ Aj+m,nfor all n, thus y is a forward orbit which
never visits Λj and which is backwards singular. Since Λj has non-empty interior
(it contains intervals), this contradicts the minimality of T , thus the forward orbit of x is not singular.
Since Tθg is minimal, every point in Bj+1must have come from Λj, thus we have
Bj+1⊆ R+◦ S+ Bj . This implies S−◦ R− B j+1⊆ S−◦ R−◦ R+◦ S+ Bj = S−◦ S+ B j = R+ Bj = Bj
for all j ≥ J; here the first two equalities use the relations in Equations (4) and the last equality follows from the definitions of Bj and R+.
Let Cj+n := S−◦ R−
n
Bj+n
. Iterating the above computation shows that
these sets are nested, Cj+n+1 ⊂ Cj+n for all n ≥ 0. The set ∩n≥0Cj+n is
non-empty since it is contained in the compact set Λj. The forward orbit of any point
in this set is either singular, or an escape orbit. The argument that such forward orbits are non-singular is identical to the one given above for the set Bj.
Finally we remark that if (s, θ) is a forward escape orbit, then (s, θ + π) is a
backward escape orbit, and vice versa.
6. Proof of density and local density of periodic points
Proof. The idea behind the proof is similar to what has been done for minimality
in Section 4. We first apply a known result to N -ringed parameters.
In this section, the direction θ varies in the proof, so we abandon the notation Tθg and we note the billiard transformation in the wind-tree by Tg(s, θ).
For each point x = (s, θ) ∈ Ωgθ and each p such that Tgp(x) exists, let us
consider the set of directions for which the orbit starting at s hits the same sequence of sides up to time p:
n
θ′: Tgi(s, θ′) and Tgi(s, θ)
lie on the same side of the same tree for i = 1, . . . , po.
This set is an open interval, we will note by θ−g(x, p), and θg+(x, p) the lower and
the upper bound of this interval. We also consider the interval (tg−, tg+) where t± is
the spatial coordinate of Tgp(s, θg ±).
Fix a N -ringed parameter f , and x = (s, θ)∈ Ωfθ,N such that (Tf)p(x) exists.
Remember that the identification between the phase space is a formal identity map discussed in Section 4. Since (Tg)p is locally continuous at x, both θg
−(x, p)
and θg+(x, p), and thus tg−(x, p) and tg+(x, p) vary continuously with respect to g
in a sufficiently small neighborhood of f . Let (sg∗, θ) := Tg
p
(s, θ), then, in a sufficiently small neighborhood of f , sg∗ varies continuously with respect to g.
By definition of tg±, for all s′
∈ (tg−, t g
+), there exists an orbit starting at (s, θ′) and
ending at (s′, θ′) for some θ′
∈ θg−(x, p), θ g +(x, p)
. Now, suppose that x = (s, θ) is Tf-periodic of period p. Note that sf
∗ = s∈ (tf−, t f
+). So there exists θ g
∗(s) such
that s, θg∗(s)is Tg-periodic and its period is a divisor of p.
Furthermore we can assume that this neighborhood V (x) of f is so small that (s, θg∗) is N1-close to (s, θ) (with respect to a fixed usual norm).
We will use the following theorem
Theorem. [BoGaKrTr, Theorem 1] In a rational polygon periodic points of the
billiard flow are dense in the phase space.
This theorem immediately implies that the same is true for the billiard map. In particular periodic points are dense in Ωfθ,N. Let{x1, . . . , xk} ⊂ Ωfθ,N be a set of
Tf-periodic points be such that{x
1, . . . , xk} is N1-dense in Ωfθ,N. Combining this
with the previous paragraph, we conclude that for every g in the neighborhood VN(f ) =TiV (xi), the set of Tg-periodic points is at least N2-dense in Ωgθ,N.
Let{fi} ⊂ AZ
2
be countable and dense, such that each fiis Ni-ringed for some
Ni. Let G := \ N≥1 [ {i:Ni≥N } VNi(fi).
ClearlyG is a dense Gδ. We have shown that every parameter inG gives rise to a
wind tree with dense periodic points.
Now additionally suppose r is rational. In this case, we can use a stronger property on periodic orbits, it is a part of a special case of Veech’s famous theorem known as the Veech dichotomy:
Theorem. [Ve, Theorem 1.4][MaTa, Theorem 5.10] If a polygon P is square tiled
then every non-singular orbit in an non-exception direction is periodic.
We will call a parameter f rationally N -ringed if f is N -ringed and all ai,j, bi,j
are rational for all (i, j)∈ EN. The key property here is that for any rationally N
-ringed parameter the N --ringed table is square-tiled and thus we can apply Veech’s theorem: there exists a countable dense set{θj} ⊂ S1such that every non-singular
point of the form (s, θj) ∈ Ωfθj,N is periodic. We call such a direction a periodic
direction.
We assume θj are enumerated so that the maximal combinatorial length of the
periodic orbits inside Ωfθj,N is increasing with j. Consider the smallest ℓ(f ) such that θ1, . . . , θℓ(f ) is N1-dense.
Let f be a rationally N -ringed parameter. Consider a periodic direction θ and the set Ωfθ,N with saddle connections removed. We decompose this set into its periodic orbit structure; more precisely this decomposition consists of a finite collection of intervals permuted by the dynamics such that the boundary of each interval from this decomposition is in a saddle connection. We call this collection of intervals D(f, θ). For each I ∈ D(f, θ), all points in I are periodic of the same period p, and we callSp−1i=0 Tfi(I) a periodic cylinder.
In the general case, we presented a construction that associates to every Tf
-periodic point x = (s, θ) and every g in a small enough neighborhood U1(s, θ) of f ,
an angle θg∗(x) such that (s, θg∗(x)) is Tg-periodic.
Because the periodic points come in cylinders, as described above for f , the angles θg∗(s, θ) and θg∗(s′, θ) will coincide for s′ in an open interval around s (if
g∈ U1(s, θ)∩ U1(s′, θ)).
For each interval I in D(f, θ), we can thus find an interval I′
⊂ I containing
at least 1− 1
ℓ(f )·N proportion of points of I such that the intersection UN(f ) :=
T
s′∈I′
U1(s′, θ) is open. For all g ∈ UN(f ) and all s, s′ ∈ I′ we have θg∗(s, θ) =
θg∗(s′, θ).
Furthermore we can assume that this neighborhood UN(f ) of f is so small that
θg∗ is N1-close to θ (with respect to a fixed usual norm).
Let{fi} ⊂ AZ
2
be countable dense and such that each fiis rationally Ni-ringed
for some Ni. Let
G := \
N≥1
[
{i:Ni≥N }
UNi(fi).
ClearlyG is a dense Gδ. We claim that every parameter inG gives rise to a wind
Figure 5. An example of periodic cylinder of length 4 (filled), and this cylinder after perturbation (striped).
an infinite subsequence (fik)⊂ (fi) such that g ∈ UNik(fik) for all k and Nik is
increasing. For sake of simplicity we denote this subsequence by (fk).
Let mkbe the measure ofSI∈D(fk,θj)I (it does not depend on j). By definition of
I′,S I∈D(fk,θj)I ′ is of measure at least 1− 1 ℓ(fk)·Nk mk. Thus Tℓ(fk) j=1 S I∈D(fk,θj)I ′ is of measure at least 1− 1 Nk
mk and thus the complement of the following infinite
measure Gδ set: \ K [ k≥K ℓ(f\k) j=1 [ I∈D(fk,θj) I′ is of zero measure. 7. Generalizations
Our results hold in a much larger framework. In the proof of minimality we only used that N -ringed configurations are dense in the space of all configurations, and that they are rational polygonal billiard tables. For the local density of periodic orbits we also used that N -ringed configurations which are Veech polygonal billiard tables are dense. Now we give some examples where these properties hold.
1) We stay in the setup discussed in the article but additionally allow the empty tree denoted by∅, thus the space of parameter is {∅} ∪ {(a, b) : r ≤ a ≤ 1 − r, r ≤ b≤ 1−r}. The Ehrenfests specifically required that the average distance A between neighboring squares is large compared to 2r. For any probability distribution m on the continuous part of the space of parameters, if we add a δ function on the empty tree, then for c < 1 large enough, the distribution cδ + (1− c)m verifies almost surely this requirement. However our result tells nothing about a full measure set of parameters for Lebesgue measure.
2) Instead of fixing r∈ [1 4,
1
2), we fix r between 0 and 1 2. If r∈ 1 2(n+1), 1 2n , place at most n2copies of trees in each cell. We can then form N -ringed configuration in
a more general sense where we replace the rhombus by an appropriate curve around the origin. One can do so using just n + 1 copies of the tree in each cell.
3) Instead of fixed size squares we use all vertical horizontal squares contained in the unit cell [0, 1]2. This set is naturally parametrized by
{t = (a, b, r) : 0 ≤ a ≤ 1, 0 ≤ b ≤ 1, 0 ≤ r ≤ min(a, b, 1 − a, 1 − b)}
where a 2r by 2r square tree is centered at the point (a, b). More generally we call a polygon a VH-tree if the sides alternate between vertical and horizontal. For example a VH-tree with 4 sides is a rectangle, with 6 sides is a figure L. We can use various subsets of VH-trees, for example all VH-trees with at most 2M sides
(M ≥ 4 fixed) contained in the unit cell. Or we can use the VH-trees with 12
sides and fixed side length r ∈ [1/4, 1/3) (called + signs). Many other interesting subclasses can be considered.
4) Fix a rational triangle P , and consider the set of all rescalings of P contained in the unit cell [0, 1]2 oriented in such a way that they have either a vertical or
horizontal side.
5) One can also change the cell structure to the hexagonal tiling and consider ap-propriate polygonal trees, for example one can use apap-propriate classes of equilateral triangular trees or hexagonal trees.
8. Acknowledgements.
We thank the anonymous referee whose detailed remarks have greatly improve the article. We gratefully acknowledge the support of ANR Perturbations and ANR GeoDyM as well as the grant APEX PAD attributed by the region PACA. This work has been carried out in the framework of the Labex Archimède (ANR-11-LABX-0033) and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the “Investissements d’Avenir” French Government program managed by the French National Research Agency (ANR)”.
Appendix A. Minimality of discontinuous maps
In this section we develop in a general context the tools we will use for proving minimality of maps with singularities.
A.1. Definitions. First, let us make precise the context in which we are using the definition of minimal map.
Definition. Let X be a locally compact metrizable topological space endowed with
a Borel measure without atoms. Let T : X 99K X be a measure-preserving map. (The dashed arrow stands for the fact that T is possibly not everywhere defined). Let us suppose that T sends homeomorphically a complement of a discrete set of points to a complement of a (possibly different) discrete set of points. We call such a map eligible.
Remark. If a map T : X 99K X is eligible there exists a setS ⊂ X that is discrete
and such that T restricted to X\ S is an homeomorphism. If S1 and S2 are two
such sets, than S1TS2 is also such a set. Indeed, S1TS2 is discrete and T is a
homeomorphism on X\ (S1TS2) = X\ S1SX\ S2.
More generally, any intersection of such sets is also such a set. Thus, there exists a minimal discrete set (w.r.t. inclusion) such that T is a homeomorphism outside this set.
Definition. Let T : X 99K X be an eligible map. We call Sing(T ) the minimal
discrete subset of X such that T restricted to X\ Sing(T ) is a homeomorphism.
We call every point in Sing(T ) a singularity of T . Remark. If T is eligible, then T−1 is also eligible and
Sing T−1= X\ T (X \ Sing(T )) .
Next, we redefine the notions of images and pre-images of sets in a way that makes clear that we will never apply the eligible transformation on its singular points.
Definition. Let T : X 99K X be an eligible map. For any A⊂ X, the image of A
by T is
T (A) :={T (x) : x ∈ A \ Sing(T )}. The preimage of A by T is its image by T−1, thus:
T−1(A) =T−1(x) : x∈ A ∩ T (X \ Sing(T )) .
We then define Tk(A) by recurrence for any integer k, as follows: Tk+1(A) =
T (Tk(A)) for any k
≥ 0; Tk−1(A) = T−1(Tk(A)) for any k
≤ 0.
Remark. The set of singularities, Sing(T ), is closed because it is discrete in a
locally compact space. Thus X\ Sing(T ) and T (X \ Sing(T )) are both open in X.
It follows that, even with this redefined notion of image and preimage, the image and preimage of an open set are always open. However, we can say nothing about the closedness of the image, or preimage, of a closed set.
Definition. Let T : X 99K X be an eligible map. Let us consider a point x0∈ X.
• The future orbit of this point is the set of all the positive iterates of T on x0, as long as T is applied to non-singular points. It is noted by O+(x0),
thus: O+(x0) := [ k≥0 Tk( {x0}) .
• In a similar way, the past orbit is O−(x 0) := [ k≤0 Tk( {x0}) .
• The orbit of a point is the union of its past and future orbit: O(x0) :=
[
k∈Z
Tk(
{x0}) .
We define the orbit of a set in a similar way. Let A⊂ X, then
• The future orbit of this set is the collection of all the positive iterates of T on A. It is noted byO+(x
0), thus:
O+(A) :={Tk(A) : k
≥ 0}. • In a similar way, the past orbit is
O−(A) :={Tk(A) : k
≤ 0}.
• The orbit of a point is the union of its past and future orbit: O−(A) :=
{Tk(A) : k
A half orbit is a future or past orbit.
We say that a point orbit or half orbit is singular if it is an orbit or half orbit of a singular point (a point in Sing(T )∪ Sing(T−1)).
Remark that the image of a non-empty set may be empty and an orbit may be finite.
Definition. Let T : X 99K X be an eligible map. A connection is a finite
non-periodic orbit. Thus, it is both an orbit of a singularity of T and an orbit of a singularity of T−1.
Remark. If x0∈ X is singular for both T and T−1, then{x0} is a connection.
Definition. Let T : X 99K X be an eligible map. We say that T is minimal if and
only if every orbit is dense.
Remark. Let T : X 99K X be an eligible map. If T is minimal and has connections,
then X is finite.
A.2. Equivalent definition of minimality.
Lemma A.1. Let T : X 99K X be an eligible map. Then T is minimal if and only
if the orbit of every open set covers X.
Proof. Suppose that T is minimal. Let U be an open set, then the orbit of
ev-ery point meets U . More precisely, for evev-ery x ∈ X, there exists k ≥ 0 such
that either x, T−1(x), . . . , T−k+1(x) are non-singular for T−1 and T−k(x)
∈ U; or x, T (x), . . . , Tk−1(x) are non-singular for T and Tk(x)
∈ U. It follows that for every x∈ X there exists an integer k such that x ∈ Tk(U ). Thus,
O(U) is an open covering of X.
Reciprocally, let us suppose that the orbit of every open set is an open covering
of X. Let us consider x ∈ X and U an open set. Because the orbit of U covers
X, there exists an integer k ≥ 0 such that x ∈ Tk(U ) or x
∈ T−k(U ). So, there
exists u∈ U such that x = Tk(u) and u, T (u), . . . , Tk−1(u) are non-singular for T ;
or x = T−k(u) and u, T−1(u), . . . , T−k+1(u) are non-singular for T−1. Thus one
has u = T−k(x) or u = Tk(x). Thus T is minimal because the orbit of any point
intersects any open set.
A.3. Keane’s minimality criterion. Keane has shown that interval exchange transformations with no connections are minimal [Ke]. Keane’s proves this fact with the usual convention that the IET is defined at singular points via left continuity. This convention does not agree with our convention that billiard orbits stop when they arrive at a corner. Thus we do not define IETs at singular points, this makes an IET an eligible map. Moreover, Keane considered IETs defined on a single interval while in our context the arising IETs are naturally defined on a finite disjoint union of intervals. More precisely:
Definition. An IET is an eligible map T : X 99K X where X ⊂ R is a finite
union of open, bounded intervals whose closures are disjoint, and T is a translation
on each connected component of X\ Sing(T ). We call T reducible if a non-trivial
finite union of connected components of X is invariant, and otherwise we call T
y x ◦ ◦ Tjy Tjx ◦ ◦ y ◦ z ◦ x ◦ Tjy ◦ Tjx ◦ Tjz ◦ Tkz ◦ Tkx ◦
Figure 6. The two ways I′ and Tj(I′) can overlap
Remark: the billiard map restricted to Ωgθ as described in the article is not an eligible map. The billiard map is defined on a disjoint union of closed intervals, if we restrict it to the interior of these intervals it becomes an eligible map. Furthermore if we restrict to the inside of a ringed it is an IET.
Keane’s result remains none the less true with a slight adjustment; for complete-ness we give a proof here.
Theorem A.2. An aperiodic, irreducible IET T : X 99K X with no connections is
minimal.
The proof of this theorem uses the following lemma (compare with [MaTa, The-orem 1.8]).
Lemma A.3. Suppose that T : X 99K X is an IET with no connections and
that O+(x) is an infinite non-periodic forward orbit. If I is an open interval with
endpoint x, thenO+(x) returns to I.
Proof. Since there are a finite number of singularities, there are a finite number of
trajectories starting at points of I that hit a singularity before crossing I again.
By shortening I to a subinterval I′ with one endpoint x we can assume that no
trajectory leaving I′ hits a singularity before returning to I′. Now consider the
forward iterates Ti(I′). By the definition of I′, these are intervals of the same
length for each i ≥ 0 until the interval returns and overlaps I′. The interval I′
must return and overlap I′ in a finite time since the total length of X is finite. Let
j be the minimum number of iterates needed until Ti(I′) overlaps I′. Tj(x)6= x
since x is not periodic.
If Tj(x) ∈ I′, we are done (see Figure 1 left). Otherwise (see Figure 1 right)
it is the trajectory leaving the other endpoint y of I′ which returns to I′ at time
j and for some z ∈ I′ we have Tj(z) = x. We now consider the interval I′′ with
endpoints z and x and apply the previous analysis to it. Orbiting I′′in the forward
direction it must return to I′′ at a certain (minimal) time k > j. We have either
Tk(z) ∈ I′′ or Tk(x) ∈ Tj(I′) But the first can not happen since it implies that
Tk(x)
∈ Tj(I′), or equivalently Tk−j(x)
∈ I′, which contradicts the minimality of
k. Thus Tk(x)
∈ I′′ as required.
Proof of Theorem A.2. By way of contradiction suppose there is a non-periodic
infinite trajectory O(x) which is not dense. Let A 6= X be the set of limit points
ofO(x). Then A is invariant under the map T . Since A 6= X and T is irreducible
one can choose a trajectoryO(y) ⊂ int(X) ∩ A \ int(A) (here the closure and the
interior are taken in R). Note thatO(y) ⊂ A since A is closed.
We will show that this trajectory is a saddle connection. We prove this by contradiction. Suppose it is not true, then O(y) is infinite in at least one of the two directions. We will show that this implies that there is an open neighborhood of y contained in A, a contradiction to y being a boundary point. Let I be an open
interval with y an endpoint. It is enough to show that there exists an open interval
(y, z) ⊂ I which is contained in A. Doing this on both sides will yield our open
neighborhood.
Lemma A.3 implies that O(y) hits I again at some point z. If the interval
(y, z)⊂ A we are done. Suppose not. Then there exists w ∈ (y, z) which is not in
A. Since A is closed, there is a largest open subinterval I′
⊂ (y, z) containing w
which is in the complement of A. Let v be the endpoint of I′ closest to y. Then,
since A is closed, v∈ A and the trajectory through v must be a saddle connection. For if it were infinite in either direction, it would intersect I′. Since A is invariant,
this contradicts that I′ misses A.
Corollary A.4. If T : X 99K X is an aperiodic and irreducible IET with no
connections, then the orbit of every open interval I covers X. Moreover, there exists K, L such thatSLk=KTk(I) = X.
Proof. The first statement is a direct corollary of Theorem A.2 and Lemma A.1.
To see that the covering happens in finite time we need to use compactness. Let I denote the closure of I in R and bI := I∪ ((X \ X) ∩ I). Then bI is open in the induced topology of bX = X.
Now consider a∈ X \ X. By the definition of IET any point b ∈ int(X) close
enough to a satisfy that: the open interval J whose endpoints are b and a is included in X, a is in bJ and there is an open interval J′⊂ X \ Sing(T ) such that T (J′) = J
(so a is in \T (J′) = bJ).
By assumptionSk∈ZTk(I) covers J′and J, thusS k∈Z
\
Tk(I) covers a. Repeating
this for all points a ∈ X \ X, it shows thatSk∈Z
\
Tk(I) is a countable open cover
of the compact set bX, and thus there exists K, L such thatSLk=KT\k(I) is a open
cover of bX. This immediately implies thatSLk=KTk(I) is an open cover of X.
A.4. Application to rational polygonal billiards. A polygon P is a compact, finitely connected, planar domain whose boundary ∂P consists of a finite union of
segments. We play billiards in P , take any point s ∈ ∂P any θ ∈ S1 such that
the vector (s, θ) points into the interior of P ; flow (s, θ) until it hits the boundary ∂P and then reflect the direction with the usual law of geometric optics, angle of incidence equals angle of reflection to produce the point (s′, θ′) = T (s, θ). T
is called the billiard map, it is not defined if s′ is a corner of the polygon. The
inverse T−1is defined if s is not a corner. A polygon is called rational if the angle
between any pair of sides is a rational multiple of π. Suppose that the angles are πmi
ni with mi and ni relatively prime; let N be the least common multiple of the
ni, and DN be the dihedral group generated by reflections in the lines through the
origin that meet at angle π
N. Let DN(θ) denote the DN orbit of a direction θ∈ S 1.
The following Theorem is a compilation of the well known results, see for example [MaTa][Sections 1.5 and 1.7]:
Theorem A.5. Suppose P is a rational polygon.
i) For each θ∈ S1 any orbit starting in the direction θ only takes directions in the
set DN(θ).
ii) For each θ ∈ S1 the restriction T
exchange transformation in the sense defined in this appendix.
iii) If θ is non-exceptional, then Tθ is irreducible and has no connections.
Combining this theorem with Corollary A.4 yields
Corollary A.6. If P is a rational polygon and θ is non-exceptional, then the map
Tθ: X→ X is minimal, and thus the orbit of every open interval I covers X.
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Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France
Address: I2M, CMI, 39 rue Joliot-Curie, F-13453 Marseille Cedex 13, France E-mail address: alba.malaga-sabogal@univ-amu.fr
E-mail address: alba.malaga@polytechnique.edu
Address: I2M, Luminy, Case 907, F-13288 Marseille CEDEX 9, France E-mail address: serge.troubetzkoy@univ-amu.fr
